Can two things be connected across any distance?

If each particle carries a definite answer to every possible measurement, arranged in advance, neither answer depends on what is chosen at the far end, and the choices are independent of the answers, then the correlations between distant measurements must satisfy an inequality. It is a constraint on that whole class of theories, and getting to it is arithmetic.

Bell did take one thing from quantum mechanics: that two measurements along the same axis always disagree. The form the first loophole-free test used, written five years later by Clauser, Horne, Shimony and Holt, takes nothing from it at all.

Quantum mechanics predicts correlations that exceed the bound. The disagreement is not philosophical. It is a number that can be measured.

Bell's 1964 paper is six pages long and was published in a journal that no longer exists. It was written as a response to an argument by Einstein, Podolsky and Rosen that quantum mechanics must be incomplete.

Their argument assumed that distant measurements cannot influence each other and concluded that the particles must carry hidden properties. Bell took that assumption seriously enough to derive its consequence, and found that it is measurable.

That is the move worth noticing: an objection that had been treated as philosophy for thirty years was turned into an inequality with a number on each side.

Early experiments violated the inequality but left two ways out. The detection loophole: if only a fraction of particles are detected, the sample might be unrepresentative. The locality loophole: if the two stations are close enough, a signal at light speed could coordinate them.

A 2015 experiment used electron spins in diamond, 1.3 kilometres apart on a university campus, with detection efficiency high enough and separation large enough to close both at once.

The inequality was violated, though not by a wide margin: 2.42 plus or minus 0.20 against a bound of 2, over 245 trials, and a p-value of 0.039 against any local realist model of space-like separated sites. Two photon experiments closed the same two loopholes later the same year, testing a related inequality with far stronger statistics: the Vienna team reported eleven and a half standard deviations.

It rules out every theory in which the outcome of each measurement is fixed in advance by properties carried locally, in which the experimenters’ choices are independent of those properties, and in which a result is final once recorded.

It does not show that information travels faster than light. The correlations cannot be used to send a message: each side alone sees random results, and the correlation only appears when the two records are brought together and compared, which happens no faster than light allows.

It also leaves open theories that keep definite properties by abandoning locality outright, and theories that deny the independence of the settings, a position with some recent advocates that no Bell test can refute outright. The 2015 experiment bounds that second one rather than leaving it alone. The violation survives a much shorter readout, which excludes any model in which the settings came into being less than 690 nanoseconds before they were recorded. A conspiracy has to be older than that. Later tests that took their settings from the light of distant quasars pushed the bound back to at least 7.8 billion years, though they had to assume fair sampling to do it.

What the equations predict is not in dispute, and the experiments agree with it. What they describe is the part that has no consensus, and the choices are set out in the question about interpretations.

That is an unusual situation for a mature science: complete agreement on the numbers and none on the picture.